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October 10, 2011 15:10 frs Sheet number 3 Page number iii cyan magenta yellow black David Henderson/Getty Images 10th EDITION CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON Drexel University IRL BIVENS Davidson College STEPHEN DAVIS Davidson College JOHN WILEY & SONS, INC.

October 10, 2011 15:10 frs Sheet number 4 Page number iv cyan magenta yellow black Publisher: Laurie Rosatone Acquisitions Editor: David Dietz Project Editor: Ellen Keohane Marketing Manager: Debi Doyle Senior Product Designer: Tom Kulesa Operations Manager: Melissa Edwards Assistant Content Editor: Beth Pearson Media Assistant Editor: Courtney Welsh Media Specialist: Laura Abrams Editorial Assistant: Elizabeth Baird, Jacqueline Sinacori Full Service Production Management: Carol Sawyer/The Perfect Proof Senior Production Editor: Kerry Weinstein Senior Designer: Madelyn Lesure Photo Editor: Sheena Goldstein Freelance Illustration: Karen Hartpence Cover Photo: David Henderson/Getty Images This book was set in LATEX by MPS Limited, a Macmillan Company, and printed and bound by R.R. Donnelley/ Jefferson City. The cover was printed by R.R. Donnelley. This book is printed on acid-free paper. Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulll their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. The paper in this book was manufactured by a mill whose forest management programs include sustained yield harvesting of its timberlands. Sustained yield harvesting principles ensure that the numbers of trees cut each year does not exceed the amount of new growth. Copyright 2012 Anton Textbooks, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 and 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, E-mail: PERMREQ@WILEY.COM. To order books or for customer service, call 1 (800)-CALL-WILEY (225-5945). ISBN 978-0-470-64769-1 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

October 10, 2011 15:10 frs Sheet number 5 Page number v cyan magenta yellow black About HOWARD ANTON Howard Anton obtained his B.A. from Lehigh University, his M.A. from the University of Illinois, and his Ph.D. from the Polytechnic University of Brooklyn, all in mathematics. In the early 1960s he worked for Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the manned space program. In 1968 he joined the Mathematics Department at Drexel University, where he taught full time until 1983. Since that time he has been an Emeritus Professor at Drexel and has devoted the majority of his time to textbook writing and activities for mathematical associations. Dr. Anton was president of the EPADEL section of the Mathematical Association of America (MAA), served on the Board of Governors of that organization, and guided the creation of the student chapters of the MAA. He has published numerous research papers in functional analysis, approximation theory, and topology, as well as pedagogical papers. He is best known for his textbooks in mathematics, which are among the most widely used in the world. There are currently more than one hundred versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French, Japanese, Chinese, Hebrew, and German. His textbook in linear algebra has won both the Textbook Excellence Award and the McGuffey Award from the Textbook Authors Association. For relaxation, Dr. Anton enjoys traveling and photography. About IRL BIVENS Irl C. Bivens, recipient of the George Polya Award and the Merten M. Hasse Prize for Expository Writing in Mathematics, received his A.B. from Pfeiffer College and his Ph.D. from the University of North Carolina at Chapel Hill, both in mathematics. Since 1982, he has taught at Davidson College, where he currently holds the position of professor of mathematics. A typical academic year sees him teaching courses in calculus, topology, and geometry. Dr. Bivens also enjoys mathematical history, and his annual History of Mathematics seminar is a perennial favorite with Davidson mathematics majors. He has published numerous articles on undergraduate mathematics, as well as research papers in his specialty, differential geometry. He has served on the editorial boards of the MAA Problem Book series, the MAA Dolciani Mathematical Expositions series and The College Mathematics Journal. When he is not pursuing mathematics, Professor Bivens enjoys reading, juggling, swimming, and walking. About STEPHEN DAVIS Stephen L. Davis received his B.A. from Lindenwood College and his Ph.D. from Rutgers University in mathematics. Having previously taught at Rutgers University and Ohio State University, Dr. Davis came to Davidson College in 1981, where he is currently a professor of mathematics. He regularly teaches calculus, linear algebra, abstract algebra, and computer science. A sabbatical in 19951996 took him to Swarthmore College as a visiting associate professor. Professor Davis has published numerous articles on calculus reform and testing, as well as research papers on nite group theory, his specialty. Professor Davis has held several ofces in the Southeastern section of the MAA, including chair and secretary-treasurer and has served on the MAA Board of Governors. He is currently a faculty consultant for the Educational Testing Service for the grading of the Advanced Placement Calculus Exam, webmaster for the North Carolina Association of Advanced Placement Mathematics Teachers, and is actively involved in nurturing mathematically talented high school students through leadership in the Charlotte Mathematics Club. For relaxation, he plays basketball, juggles, and travels. Professor Davis and his wife Elisabeth have three children, Laura, Anne, and James, all former calculus students.

October 10, 2011 15:10 frs Sheet number 6 Page number vi cyan magenta yellow black To my wife Pat and my children: Brian, David, and Lauren In Memory of my mother Shirley my father Benjamin my thesis advisor and inspiration, George Bachman my benefactor in my time of need, Stephen Girard (17501831) HA To my son Robert IB To my wife Elisabeth my children: Laura, Anne, and James SD

September 30, 2011 17:46 fpref Sheet number 1 Page number vii cyan magenta yellow black PREFACE This tenth edition of Calculus maintains those aspects of previous editions that have led to the series successwe continue to strive for student comprehension without sacricing mathematical accuracy, and the exercise sets are carefully constructed to avoid unhappy surprises that can derail a calculus class. All of the changes to the tenth edition were carefully reviewed by outstanding teachers comprised of both users and nonusers of the previous edition. The charge of this committee was to ensure that all changes did not alter those aspects of the text that attracted users of the ninth edition and at the same time provide freshness to the new edition that would attract new users. NEW TO THIS EDITION Exercise sets have been modied to correspond more closely to questions in WileyPLUS. In addition, more WileyPLUS questions now correspond to specic exercises in the text. New applied exercises have been added to the book and existing applied exercises have been updated. Where appropriate, additional skill/practice exercises were added. OTHER FEATURES Flexibility Thiseditionhasabuilt-inexibilitythatisdesignedtoserveabroadspectrum of calculus philosophiesfrom traditional to reform. Technology can be emphasized or not, and the order of many topics can be permuted freely to accommodate each instructors specic needs. Rigor The challenge of writing a good calculus book is to strike the right balance between rigor and clarity. Our goal is to present precise mathematics to the fullest extent possible in an introductory treatment. Where clarity and rigor conict, we choose clarity; however, we believe it to be important that the student understand the difference between a careful proof and an informal argument, so we have informed the reader when the arguments being presented are informal or motivational. Theory involving - arguments appears in separate sections so that they can be covered or not, as preferred by the instructor. Rule of Four The rule of four refers to presenting concepts from the verbal, algebraic, visual, and numerical points of view. In keeping with current pedagogical philosophy, we used this approach whenever appropriate. Visualization This edition makes extensive use of modern computer graphics to clarify concepts and to develop the students ability to visualize mathematical objects, particularly those in 3-space. For those students who are working with graphing technology, there are vii

September 30, 2011 17:46 fpref Sheet number 2 Page number viii cyan magenta yellow black viii Preface many exercises that are designed to develop the students ability to generate and analyze mathematical curves and surfaces. Quick Check Exercises Each exercise set begins with approximately ve exercises (answers included) that are designed to provide students with an immediate assessment of whether they have mastered key ideas from the section. They require a minimum of computation and are answered by lling in the blanks. Focus on Concepts Exercises Each exercise set contains a clearly identied group of problems that focus on the main ideas of the section. Technology Exercises Most sections include exercises that are designed to be solved using either a graphing calculator or a computer algebra system such as Mathematica, Maple, or the open source program Sage. These exercises are marked with an icon for easy identication. Applicability of Calculus One of the primary goals of this text is to link calculus to the real world and the students own experience. This theme is carried through in the examples and exercises. Career Preparation This text is written at a mathematical level that will prepare stu- dents for a wide variety of careers that require a sound mathematics background, including engineering, the various sciences, and business. Trigonometry Review Deciencies in trigonometry plague many students, so we have included a substantial trigonometry review in Appendix B. Appendix on Polynomial Equations Because many calculus students are weak in solving polynomial equations, we have included an appendix (Appendix C) that reviews the Factor Theorem, the Remainder Theorem, and procedures for nding rational roots. Principles of Integral Evaluation The traditional Techniques of Integration is entitled Principles of Integral Evaluation to reect its more modern approach to the material. The chapter emphasizes general methods and the role of technology rather than specic tricks for evaluating complicated or obscure integrals. Historical Notes The biographies and historical notes have been a hallmark of this text from its rst edition and have been maintained. All of the biographical materials have been distilled from standard sources with the goal of capturing and bringing to life for the student the personalities of historys greatest mathematicians. Margin Notes and Warnings These appear in the margins throughout the text to clarify or expand on the text exposition or to alert the reader to some pitfall.

October 4, 2011 14:06 f-sup Sheet number 1 Page number ix cyan magenta yellow black SUPPLEMENTS The Student Solutions Manual, which is printed in two volumes, provides detailed solu- tions to the odd-numbered exercises in the text. The structure of the step-by-step solutions matches those of the worked examples in the textbook. The Student Solutions Manual is also provided in digital format to students in WileyPLUS. Volume I (Single-Variable Calculus, Early Transcendentals) ISBN: 978-1-118-17381-7 Volume II (Multivariable Calculus, Early Transcendentals) ISBN: 978-1-118-17383-1 The Student Study Guide is available for download from the book companion Web site at www.wiley.com/college/anton or at www.howardanton.com and to users of WileyPLUS. The Instructors Solutions Manual, which is printed in two volumes, contains detailed solutionstoalloftheexercisesinthetext. TheInstructorsSolutionsManualisalsoavailable in PDF format on the password-protected Instructor Companion Site at www.wiley.com/ college/anton or at www.howardanton.com and in WileyPLUS. Volume I (Single-Variable Calculus, Early Transcendentals) ISBN: 978-1-118-17378-7 Volume II (Multivariable Calculus, Early Transcendentals) ISBN: 978-1-118-17379-4 The Instructors Manual suggests time allocations and teaching plans for each section in the text. Most of the teaching plans contain a bulleted list of key points to emphasize. The discussion of each section concludes with a sample homework assignment. The Instructors Manual is available in PDF format on the password-protected Instructor Companion Site at www.wiley.com/college/anton or at www.howardanton.com and in WileyPLUS. The Web Projects (Expanding the Calculus Horizon) referenced in the text can also be downloaded from the companion Web sites and from WileyPLUS. Instructors can also access the following materials from the book companion site or WileyPLUS: Interactive Illustrations can be used in the classroom or computer lab to present and explore key ideas graphically and dynamically. They are especially useful for display of three-dimensional graphs in multivariable calculus. The Computerized Test Bank features more than 4000 questionsmostly algorithmi- cally generatedthat allow for varied questions and numerical inputs. The Printable Test Bank features questions and answers for every section of the text. PowerPoint lecture slides cover the major concepts and themes of each section of the book. Personal-Response System questions (Clicker Questions) appear at the end of each PowerPoint presentation and provide an easy way to gauge classroom understanding. Additional calculus content covers analytic geometry in calculus, mathematical mod- eling with differential equations and parametric equations, as well as an introduction to linear algebra. ix

October 4, 2011 14:06 f-sup Sheet number 2 Page number x cyan magenta yellow black x Supplements WileyPLUS WileyPLUS, Wileys digital-learning environment, is loaded with all of the supplements listed on the previous page, and also features the following: Homework management tools, which easily allow you to assign and grade algorithmic questions, as well as gauge student comprehension. Algorithmic questions with randomized numeric values and an answer-entry palette for symbolic notation are provided online though WileyPLUS. Students can click on help buttons for hints, link to the relevant section of the text, show their work or query their instructor using a white board, or see a step-by-step solution (depending on instructor- selecting settings). Interactive Illustrations can be used in the classroom or computer lab, or for student practice. QuickStart predesigned reading and homework assignments. Use them as-is or customize them to t the needs of your classroom. The e-book, which is an exact version of the print text but also features hyperlinks to questions, denitions, and supplements for quicker and easier support. Guided Online (GO) Tutorial Exercises that prompt students to build solutions step by step. Rather than simply grading an exercise answer as wrong, GO tutorial problems show students precisely where they are making a mistake. AreYou Ready? quizzes gauge student mastery of chapter concepts and techniques and provide feedback on areas that require further attention. Algebra and Trigonometry Refresher quizzes provide students with an opportunity to brush up on the material necessary to master calculus, as well as to determine areas that require further review. WileyPLUS. Learn more at www.wileyplus.com.

September 30, 2011 21:00 back Sheet number 1 Page number xi cyan magenta yellow black ACKNOWLEDGMENTS It has been our good fortune to have the advice and guidance of many talented people whose knowledge and skills have enhanced this book in many ways. For their valuable help we thank the following people. Reviewers of the Tenth Edition Frederick Adkins, Indiana University of Pennsylvania Gerardo Aladro, Florida International University Mike Albanese, Central Piedmont Community College Faiz Al-Rubaee, University of North Florida Mahboub Baccouch, University of Nebraska at Omaha Jim Brandt, Southern Utah University Elizabeth Brown, James Madison University Michael Brown, San Diego Mesa College Christopher Butler, Case Western Reserve University Nick Bykov, San Joaquin Delta College Jamylle Carter, Diablo Valley College Hongwei Chen, Christopher Newport University David A. Clark, Randolph-Macon College Dominic P. Clemence, North Carolina Agricultural and Technical State University Michael Cohen, Hofstra University Hugh Cornell, Salt Lake Community College Kyle Costello, Salt Lake Community College Walter Czarnec, Framingham State University Michael Daniel, Drexel University Judith Downey, University of Nebraska, Omaha Artur Elezi, American University David James Ellingson, Napa Valley College Elaine B. Fitt, Bucks County Community College Greg Gibson, North Carolina Agricultural and Technical State University Yvonne A. Greenbaun, Mercer County Community College Jerome I. Heaven, Indiana Tech Kathryn Lesh, Union College Eric Matsuoka, Leeward Community College Ted Nirgiotis, Diablo Valley College Mihaela Poplicher, University of Cincinnati Adrian R. Ranic, Erie Community CollegeNorth Thomas C. Redd, North Carolina Agricultural and Technical State University R. A. Rock, Daniel Webster College John Paul Roop, North Carolina Agricultural and Technical State University Philippe Rukimbira, Florida International University Dee Dee Shaulis, University of Colorado at Boulder Michael D. Shaw, Florida Institute of Technology Jennifer Siegel, Broward CollegeCentral Campus Thomas W. Simpson, University of South Carolina Union Maria Siopsis, Maryville College Mark A. Smith, Miami University, Ohio Alan Taylor, Union College Kathy Vranicar, University of Nebraska, Omaha Anke Walz, Kutztown University Zhi-Qiang Wang, Utah State University Tom Wells, Delta College Greg Wisloski, Indiana University of Pennsylvania Reviewers and Contributors to the Ninth Edition Frederick Adkins, Indiana University of Pennsylvania Bill Allen, Reedley College-Clovis Center Jerry Allison, Black Hawk College Seth Armstrong, Southern Utah University Przemyslaw Bogacki, Old Dominion University David Bradley, University of Maine Wayne P. Britt, Louisiana State University Dean Burbank, Gulf Coast Community College Jason Cantarella, University of Georgia Yanzhao Cao, Florida A&M University Kristin Chatas, Washtenaw Community College Michele Clement, Louisiana State University Ray Collings, Georgia Perimeter College David E. Dobbs, University of Tennessee, Knoxville H. Edward Donley, Indiana University of Pennsylvania T. J. Duda, Columbus State Community College Jim Edmondson, Santa Barbara City College Nancy Eschen, Florida Community College, Jacksonville Reuben Farley, Virginia Commonwealth University Michael Filaseta, University of South Carolina Jose Flores, University of South Dakota Mitch Francis, Horace Mann Berit N. Givens, California State Polytechnic University, Pomona Zhuang-dan Guan, University of California, Riverside Jerome Heaven, Indiana Tech Greg Henderson, Hillsborough Community College Patricia Henry, Drexel University Danrun Huang, St. Cloud State University Alvaro Islas, University of Central Florida Micah James, University of Illinois xi

September 30, 2011 21:00 back Sheet number 2 Page number xii cyan magenta yellow black xii Acknowledgments Bin Jiang, Portland State University Ronald Jorgensen, Milwaukee School of Engineering Mohammad Kazemi, University of North Carolina, Charlotte Raja Khoury, Collin County Community College Przemo Kranz, University of Mississippi Carole King Krueger, The University of Texas at Arlington Steffen Lempp, University of Wisconsin, Madison Thomas Leness, Florida International University Kathryn Lesh, Union College Wen-Xiu Ma, University of South Florida Behailu Mammo, Hofstra University Vania Mascioni, Ball State University John McCuan, Georgia Tech Daryl McGinnis, Columbus State Community College Michael Mears, Manatee Community College John G. Michaels, SUNY Brockport Jason Miner, Santa Barbara City College Darrell Minor, Columbus State Community College Kathleen Miranda, SUNY Old Westbury Carla Monticelli, Camden County College Bryan Mosher, University of Minnesota Ferdinand O. Orock, Hudson County Community College Altay Ozgener, Manatee Community College Chuang Peng, Morehouse College Joni B. Pirnot, Manatee Community College Elise Price, Tarrant County College David Price, Tarrant County College Holly Puterbaugh, University of Vermont Hah Suey Quan, Golden West College Joseph W. Rody, Arizona State University Jan Rychtar, University of North Carolina, Greensboro John T. Saccoman, Seton Hall University Constance Schober, University of Central Florida Kurt Sebastian, United States Coast Guard Paul Seeburger, Monroe Community College Charlotte Simmons, University of Central Oklahoma Don Soash, Hillsborough Community College Bradley Stetson, Schoolcraft College Bryan Stewart, Tarrant County College Walter E. Stone, Jr., North Shore Community College Eleanor Storey, Front Range Community College, Westminster Campus Stefania Tracogna, Arizona State University Helene Tyler, Manhattan College Pavlos Tzermias, University of Tennessee, Knoxville Raja Varatharajah, North Carolina Agricultural and Technical State University Francis J. Vasko, Kutztown University David Voss, Western Illinois University Jim Voss, Front Range Community College Anke Walz, Kutztown Community College Richard Watkins, Tidewater Community College Xian Wu, University of South Carolina Yvonne Yaz, Milwaukee School of Engineering Richard A. Zang, University of New Hampshire Xiao-Dong Zhang, Florida Atlantic University Diane Zych, Erie Community College We would also like to thank Celeste Hernandez and Roger Lipsett for their accuracy check of the tenth edition. Thanks also go to Tamas Wiandt for revising the solutions manuals, and Przemyslaw Bogacki for accuracy checking those solutions; Brian Camp and Lyle Smith for their revision of the Student Study Guide; Jim Hartman for his revision of the Instructors Manual; Ann Ostberg for revising the PowerPoint slides; Beverly Fuseld for creating new GO Tutorials, and Mark McKibben for accuracy checking these new tutorials. We also appreciate the feedback we received from Mark Dunster, Cecelia Knoll, and Michael Rosenthal on selected WileyPLUS problems.

September 30, 2011 18:19 ftoc Sheet number 1 Page number xiii cyan magenta yellow black CONTENTS 0 BEFORE CALCULUS 1 0.1 Functions 1 0.2 New Functions from Old 15 0.3 Families of Functions 27 0.4 Inverse Functions; Inverse Trigonometric Functions 38 0.5 Exponential and Logarithmic Functions 52 1 LIMITS AND CONTINUITY 67 1.1 Limits (An Intuitive Approach) 67 1.2 Computing Limits 80 1.3 Limits at Innity; End Behavior of a Function 89 1.4 Limits (Discussed More Rigorously) 100 1.5 Continuity 110 1.6 Continuity of Trigonometric, Exponential, and Inverse Functions 121 2 THE DERIVATIVE 131 2.1 Tangent Lines and Rates of Change 131 2.2 The Derivative Function 143 2.3 Introduction to Techniques of Differentiation 155 2.4 The Product and Quotient Rules 163 2.5 Derivatives of Trigonometric Functions 169 2.6 The Chain Rule 174 3 TOPICS IN DIFFERENTIATION 185 3.1 Implicit Differentiation 185 3.2 Derivatives of Logarithmic Functions 192 3.3 Derivatives of Exponential and Inverse Trigonometric Functions 197 3.4 Related Rates 204 3.5 Local Linear Approximation; Differentials 212 3.6 LHpitals Rule; Indeterminate Forms 219 xiii

September 30, 2011 18:19 ftoc Sheet number 2 Page number xiv cyan magenta yellow black xiv Contents 4 THE DERIVATIVE IN GRAPHING AND APPLICATIONS 232 4.1 Analysis of Functions I: Increase, Decrease, and Concavity 232 4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 244 4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 254 4.4 Absolute Maxima and Minima 266 4.5 Applied Maximum and Minimum Problems 274 4.6 Rectilinear Motion 288 4.7 Newtons Method 296 4.8 Rolles Theorem; Mean-Value Theorem 302 5 INTEGRATION 316 5.1 An Overview of the Area Problem 316 5.2 The Indenite Integral 322 5.3 Integration by Substitution 332 5.4 The Denition of Area as a Limit; Sigma Notation 340 5.5 The Denite Integral 353 5.6 The Fundamental Theorem of Calculus 362 5.7 Rectilinear Motion Revisited Using Integration 376 5.8 Average Value of a Function and its Applications 385 5.9 Evaluating Denite Integrals by Substitution 390 5.10 Logarithmic and Other Functions Dened by Integrals 396 6 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 413 6.1 Area Between Two Curves 413 6.2 Volumes by Slicing; Disks and Washers 421 6.3 Volumes by Cylindrical Shells 432 6.4 Length of a Plane Curve 438 6.5 Area of a Surface of Revolution 444 6.6 Work 449 6.7 Moments, Centers of Gravity, and Centroids 458 6.8 Fluid Pressure and Force 467 6.9 Hyperbolic Functions and Hanging Cables 474 7 PRINCIPLES OF INTEGRAL EVALUATION 488 7.1 An Overview of Integration Methods 488 7.2 Integration by Parts 491 7.3 Integrating Trigonometric Functions 500 7.4 Trigonometric Substitutions 508 7.5 Integrating Rational Functions by Partial Fractions 514 7.6 Using Computer Algebra Systems and Tables of Integrals 523

September 30, 2011 18:19 ftoc Sheet number 3 Page number xv cyan magenta yellow black Contents xv 7.7 Numerical Integration; Simpsons Rule 533 7.8 Improper Integrals 547 8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 561 8.1 Modeling with Differential Equations 561 8.2 Separation of Variables 568 8.3 Slope Fields; Eulers Method 579 8.4 First-Order Differential Equations and Applications 586 9 INFINITE SERIES 596 9.1 Sequences 596 9.2 Monotone Sequences 607 9.3 Innite Series 614 9.4 Convergence Tests 623 9.5 The Comparison, Ratio, and Root Tests 631 9.6 Alternating Series; Absolute and Conditional Convergence 638 9.7 Maclaurin and Taylor Polynomials 648 9.8 Maclaurin and Taylor Series; Power Series 659 9.9 Convergence of Taylor Series 668 9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 678 10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS 692 10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 692 10.2 Polar Coordinates 705 10.3 Tangent Lines, Arc Length, and Area for Polar Curves 719 10.4 Conic Sections 730 10.5 Rotation of Axes; Second-Degree Equations 748 10.6 Conic Sections in Polar Coordinates 754 11 THREE-DIMENSIONAL SPACE; VECTORS 767 11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 767 11.2 Vectors 773 11.3 Dot Product; Projections 785 11.4 Cross Product 795 11.5 Parametric Equations of Lines 805 11.6 Planes in 3-Space 813 11.7 Quadric Surfaces 821 11.8 Cylindrical and Spherical Coordinates 832

September 30, 2011 18:19 ftoc Sheet number 4 Page number xvi cyan magenta yellow black xvi Contents 12 VECTOR-VALUED FUNCTIONS 841 12.1 Introduction to Vector-Valued Functions 841 12.2 Calculus of Vector-Valued Functions 848 12.3 Change of Parameter; Arc Length 858 12.4 Unit Tangent, Normal, and Binormal Vectors 868 12.5 Curvature 873 12.6 Motion Along a Curve 882 12.7 Keplers Laws of Planetary Motion 895 13 PARTIAL DERIVATIVES 906 13.1 Functions of Two or More Variables 906 13.2 Limits and Continuity 917 13.3 Partial Derivatives 927 13.4 Differentiability, Differentials, and Local Linearity 940 13.5 The Chain Rule 949 13.6 Directional Derivatives and Gradients 960 13.7 Tangent Planes and Normal Vectors 971 13.8 Maxima and Minima of Functions of Two Variables 977 13.9 Lagrange Multipliers 989 14 MULTIPLE INTEGRALS 1000 14.1 Double Integrals 1000 14.2 Double Integrals over Nonrectangular Regions 1009 14.3 Double Integrals in Polar Coordinates 1018 14.4 Surface Area; Parametric Surfaces 1026 14.5 Triple Integrals 1039 14.6 Triple Integrals in Cylindrical and Spherical Coordinates 1048 14.7 Change of Variables in Multiple Integrals; Jacobians 1058 14.8 Centers of Gravity Using Multiple Integrals 1071 15 TOPICS IN VECTOR CALCULUS 1084 15.1 Vector Fields 1084 15.2 Line Integrals 1094 15.3 Independence of Path; Conservative Vector Fields 1111 15.4 Greens Theorem 1122 15.5 Surface Integrals 1130 15.6 Applications of Surface Integrals; Flux 1138 15.7 The Divergence Theorem 1148 15.8 Stokes Theorem 1158

September 30, 2011 18:19 ftoc Sheet number 5 Page number xvii cyan magenta yellow black Contents xvii A APPENDICES A GRAPHING FUNCTIONS USING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS A1 B TRIGONOMETRY REVIEW A13 C SOLVING POLYNOMIAL EQUATIONS A27 D SELECTED PROOFS A34 ANSWERS TO ODD-NUMBERED EXERCISES A45 INDEX I-1 WEB APPENDICES (online only) Available for download at www.wiley.com/college/anton or at www.howardanton.com and in WileyPLUS. E REAL NUMBERS, INTERVALS, AND INEQUALITIES F ABSOLUTE VALUE G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS H DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS I EARLY PARAMETRIC EQUATIONS OPTION J MATHEMATICAL MODELS K THE DISCRIMINANT L SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS WEB PROJECTS: Expanding the Calculus Horizon (online only) Available for download at www.wiley.com/college/anton or at www.howardanton.com and in WileyPLUS. BLAMMO THE HUMAN CANNONBALL COMET COLLISION HURRICANE MODELING ITERATION AND DYNAMICAL SYSTEMS RAILROAD DESIGN ROBOTICS

October 15, 2011 13:47 ast Sheet number 1 Page number xviii cyan magenta yellow black xviii The Roots of Calculus THE ROOTS OF CALCULUS Todays exciting applications of calculus have roots that can be traced to the work of the Greek mathematicianArchimedes, but the actual discovery of the fundamental principles of cal- culus was made independently by Isaac Newton (English) and Gottfried Leibniz (German) in the late seventeenth century. The work of Newton and Leibniz was motivated by four major classes of scientic and mathematical problems of the time: Find the tangent line to a general curve at a given point. Find the area of a general region, the length of a general curve, and the volume of a general solid. Find the maximum or minimum value of a quantityfor example, the maximum and minimum distances of a planet from the Sun, or the maximum range attainable for a pro- jectile by varying its angle of re. Given a formula for the distance traveled by a body in any specied amount of time, nd the velocity and acceleration of the body at any instant. Conversely, given a formula that species the acceleration of velocity at any instant, nd the distance traveled by the body in a specied period of time. Newton and Leibniz found a fundamental relationship be- tween the problem of nding a tangent line to a curve and the problem of determining the area of a region. Their real- ization of this connection is considered to be the discovery of calculus. Though Newton saw how these two problems are related ten years before Leibniz did, Leibniz published his work twenty years before Newton. This situation led to a stormy debate over who was the rightful discoverer of calculus. The debate engulfed Europe for half a century, with the scien- tists of the European continent supporting Leibniz and those from England supporting Newton. The conict was extremely unfortunate because Newtons inferior notation badly ham- pered scientic development in England, and the Continent in turn lost the benet of Newtons discoveries in astronomy and physics for nearly fty years. In spite of it all, Newton and Leibniz were sincere admirers of each others work. [Image: Public domain image from http://commons.wikimedia.org/ wiki/File:Hw-newton.jpg. Image provided courtesy of the University of Texas Libraries, The University of Texas at Austin.] ISAAC NEWTON (16421727) Newton was born in the village of Woolsthorpe, England. His father died before he was born and his mother raised him on the family farm. As a youth he showed little evidence of his later brilliance, except for an unusual talent with mechanical deviceshe apparently built a working water clock and a toy our mill powered by a mouse. In 1661 he entered Trinity College in Cambridge with a deciency in geometry. Fortunately, Newton caught the eye of Isaac Barrow, a gifted mathematician and teacher. Under Barrows guidance Newton immersed himself in mathematics and science, but he graduated without any special distinction. Because the bubonic plague was spreading rapidly through London, Newton returned to his home in Woolsthorpe and stayed there during the years of 1665 and 1666. In those two momentous years the entire framework of modern science was miraculously created in Newtons mind. He discovered calculus, recognized the underlying principles of planetary motion and gravity, and determined that white sunlight was composed of all colors, red to violet. For whatever reasons he kept his discoveries to himself. In 1667 he returned to Cambridge to obtain his Masters degree and upon graduation became a teacher at Trinity. Then in 1669 Newton succeeded his teacher, Isaac Barrow, to the Lucasian chair of mathematics at Trinity, one of the most honored chairs of mathematics in the world. Thereafter, brilliantdiscoveriesowedfromNewtonsteadily. Heformulated the law of gravitation and used it to explain the motion of the moon, the planets, and the tides; he formulated basic theories of light, thermodynamics, and hydrodynamics; and he devised and constructed the rst modern reecting telescope. Throughout his life Newton was hesitant to publish his major discoveries, revealing them only to a select

October 15, 2011 13:09 fmend Sheet number 1 Page number 2 cyan magenta yellow black GEOMETRY FORMULAS A = area, S = lateral surface area, V = volume, h = height, B = area of base, r = radius, l = slant height, C = circumference, s = arc length V = Bh h B h b A = bh h b A = bh1 2 1 3 A = (a + b)h1 2 h b a A = pr2 , C = 2pr r A = r2 u, s = ru (u in radians) r s u 1 2 h r V = pr2 h, S = 2prh V = pr3 , S = 4pr2 r 4 3 l r h V = pr2 h, S = prl h B Parallelogram Right Circular Cylinder Right Circular Cone Any Cylinder or Prism with Parallel Bases Sphere Triangle Trapezoid Circle Sector ALGEBRA FORMULAS THE QUADRATIC FORMULA THE BINOMIAL FORMULA The solutions of the quadratic equation ax2 + bx + c = 0 are x = b b2 4ac 2a (x + y)n = xn + nxn1 y + n(n 1) 1 2 xn2 y2 + n(n 1)(n 2) 1 2 3 xn3 y3 + + nxyn1 + yn (x y)n = xn nxn1 y + n(n 1) 1 2 xn2 y2 n(n 1)(n 2) 1 2 3 xn3 y3 + nxyn1 yn TABLE OF INTEGRALS BASIC FUNCTIONS 1. un du = un+1 n + 1 + C 2. du u = ln |u| + C 3. eu du = eu + C 4. sin u du = cos u + C 5. cos u du = sin u + C 6. tan u du = ln |sec u| + C 7. sin1 u du = u sin1 u + 1 u2 + C 8. cos1 u du = u cos1 u 1 u2 + C 9. tan1 u du = u tan1 u ln 1 + u2 + C 10. au du = au ln a + C 11. ln u du = u ln u u + C 12. cot u du = ln |sin u| + C 13. sec u du = ln |sec u + tan u| + C = ln |tan 1 4 + 1 2 u | + C 14. csc u du = ln |csc u cot u| + C = ln |tan 1 2 u| + C 15. cot1 u du = u cot1 u + ln 1 + u2 + C 16. sec1 u du = u sec1 u ln |u + u2 1| + C 17. csc1 u du = u csc1 u + ln |u + u2 1| + C

October 15, 2011 13:09 fmend Sheet number 2 Page number 3 cyan magenta yellow black RECIPROCALS OF BASIC FUNCTIONS 18. 1 1 sin u du = tan u sec u + C 19. 1 1 cos u du = cot u csc u + C 20. 1 1 tan u du = 1 2 (u ln |cos u sin u|) + C 21. 1 sin u cos u du = ln |tan u| + C 22. 1 1 cot u du = 1 2 (u ln |sin u cos u|) + C 23. 1 1 sec u du = u + cot u csc u + C 24. 1 1 csc u du = u tan u sec u + C 25. 1 1 eu du = u ln(1 eu ) + C POWERS OF TRIGONOMETRIC FUNCTIONS 26. sin2 u du = 1 2 u 1 4 sin 2u + C 27. cos2 u du = 1 2 u + 1 4 sin 2u + C 28. tan2 u du = tan u u + C 29. sinn u du = 1 n sinn1 u cos u + n 1 n sinn2 u du 30. cosn u du = 1 n cosn1 u sin u + n 1 n cosn2 u du 31. tann u du = 1 n 1 tann1 u tann2 u du 32. cot2 u du = cot u u + C 33. sec2 u du = tan u + C 34. csc2 u du = cot u + C 35. cotn u du = 1 n 1 cotn1 u cotn2 u du 36. secn u du = 1 n 1 secn2 u tan u + n 2 n 1 secn2 u du 37. cscn u du = 1 n 1 cscn2 u cot u + n 2 n 1 cscn2 u du PRODUCTS OF TRIGONOMETRIC FUNCTIONS 38. sin mu sin nu du = sin(m + n)u 2(m + n) + sin(m n)u 2(m n) + C 39. cos mu cos nu du = sin(m + n)u 2(m + n) + sin(m n)u 2(m n) + C 40. sin mu cos nu du = cos(m + n)u 2(m + n) cos(m n)u 2(m n) + C 41. sinm u cosn u du = sinm1 u cosn+1 u m + n + m 1 m + n sinm2 u cosn u du = sinm+1 u cosn1 u m + n + n 1 m + n sinm u cosn2 u du PRODUCTS OF TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 42. eau sin bu du = eau a2 + b2 (a sin bu b cos bu) + C 43. eau cos bu du = eau a2 + b2 (a cos bu + b sin bu) + C POWERS OF u MULTIPLYING OR DIVIDING BASIC FUNCTIONS 44. u sin u du = sin u u cos u + C 45. u cos u du = cos u + u sin u + C 46. u2 sin u du = 2u sin u + (2 u2 ) cos u + C 47. u2 cos u du = 2u cos u + (u2 2) sin u + C 48. un sin u du = un cos u + n un1 cos u du 49. un cos u du = un sin u n un1 sin u du 50. un ln u du = un+1 (n + 1)2 [(n + 1) ln u 1] + C 51. ueu du = eu (u 1) + C 52. un eu du = un eu n un1 eu du 53. un au du = unau ln a n ln a un1 au du + C 54. eu du un = eu (n 1)un1 + 1 n 1 eu du un1 55. au du un = au (n 1)un1 + ln a n 1 au du un1 56. du u ln u = ln |ln u| + C POLYNOMIALS MULTIPLYING BASIC FUNCTIONS 57. p(u)eau du = 1 a p(u)eau 1 a2 p (u)eau + 1 a3 p (u)eau [signs alternate: + + ] 58. p(u) sin au du = 1 a p(u) cos au + 1 a2 p (u) sin au + 1 a3 p (u) cos au [signs alternate in pairs after rst term: + + + + ] 59. p(u) cos au du = 1 a p(u) sin au + 1 a2 p (u) cos au 1 a3 p (u) sin au [signs alternate in pairs: + + + + ]

October 28, 2011 17:19 for-the-student Sheet number 1 Page number ii cyan magenta yellow black FOR THE STUDENT Calculus provides a way of viewing and analyzing the physi- cal world. As with all mathematics courses, calculus involves equations and formulas. However, if you successfully learn to use all the formulas and solve all of the problems in the text but do not master the underlying ideas, you will have missed the most important part of calculus. If you master these ideas, you will have a widely applicable tool that goes far beyond textbook exercises. Before starting your studies, you may nd it helpful to leaf through this text to get a general feeling for its different parts: The opening page of each chapter gives you an overview of what that chapter is about, and the opening page of each section within a chapter gives you an overview of what that section is about. To help you locate specic information, sections are subdivided into topics that are marked with a box like this . Each section ends with a set of exercises. The answers to most odd-numbered exercises appear in the back of the book. If you nd that your answer to an exercise does not match that in the back of the book, do not assume immedi- ately that yours is incorrectthere may be more than one way to express the answer. For example, if your answer is 2/2 and the text answer is 1/ 2 , then both are correct since your answer can be obtained by rationalizing the text answer. In general, if your answer does not match that in the text, then your best rst step is to look for an algebraic manipulation or a trigonometric identity that might help you determine if the two answers are equivalent. If the answer is in the form of a decimal approximation, then your answer might differ from that in the text because of a difference in the number of decimal places used in the computations. The section exercises include regular exercises and four special categories: Quick Check, Focus on Concepts, True/False, and Writing. TheQuickCheck exercisesareintendedtogiveyouquick feedback on whether you understand the key ideas in the section; they involve relatively little computation, and have answers provided at the end of the exercise set. The Focus on Concepts exercises, as their name suggests, key in on the main ideas in the section. True/False exercises focus on key ideas in a different way. You must decide whether the statement is true in all possible circumstances, in which case you would declare it to be true, or whether there are some circumstances in which it is not true, in which case you would declare it to be false. In each such exercise you are asked to Explain your answer. You might do this by noting a theorem in the text that shows the statement to be true or by nding a particular example in which the statement is not true. Writing exercises are intended to test your ability to ex- plain mathematical ideas in words rather than relying solely on numbers and symbols. All exercises requiring writing should be answered in complete, correctly punc- tuated logical sentencesnot with fragmented phrases and formulas. Each chapter ends with two additional sets of exercises: Chapter Review Exercises, which, as the name suggests, is a select set of exercises that provide a review of the main concepts and techniques in the chapter, and Making Con- nections, in which exercises require you to draw on and combine various ideas developed throughout the chapter. Your instructor may choose to incorporate technology in your calculus course. Exercises whose solution involves the use of some kind of technology are tagged with icons to alert you and your instructor. Those exercises tagged with the icon require graphing technologyeither a graphing calculator or a computer program that can graph equations. Those exercises tagged with the icon C require a com- puter algebra system (CAS) such as Mathematica, Maple, or available on some graphing calculators. At the end of the text you will nd a set of four appen- dices covering various topics such as a detailed review of trigonometry and graphing techniques using technology. Inside the front and back covers of the text you will nd endpapers that contain useful formulas. The ideas in this text were created by real people with in- teresting personalities and backgrounds. Pictures and bio- graphical sketches of many of these people appear through- out the book. Notes in the margin are intended to clarify or comment on important points in the text. A Word of Encouragement As you work your way through this text you will nd some ideas that you understand immediately, some that you dont understand until you have read them several times, and others that you do not seem to understand, even after several readings. Do not become discouragedsome ideas are intrinsically dif- cult and take time to percolate. You may well nd that a hard idea becomes clear later when you least expect it. Web Sites for this Text www.antontextbooks.com www.wiley.com/go/global/anton

August 31, 2011 19:31 C00 Sheet number 1 Page number 1 cyan magenta yellow black 1 Arco Images/Alamy 0 The development of calculus in the seventeenth and eighteenth centuries was motivated by the need to understand physical phenomena such as the tides, the phases of the moon, the nature of light, and gravity. One of the important themes in calculus is the analysis of relationships between physical or mathematical quantities. Such relationships can be described in terms of graphs, formulas, numerical data, or words. In this chapter we will develop the concept of a function, which is the basic idea that underlies almost all mathematical and physical relationships, regardless of the form in which they are expressed. We will study properties of some of the most basic functions that occur in calculus, including polynomials, trigonometric functions, inverse trigonometric functions, exponential functions, and logarithmic functions. BEFORE CALCULUS 0.1 FUNCTIONS In this section we will dene and develop the concept of a function, which is the basic mathematical object that scientists and mathematicians use to describe relationships between variable quantities. Functions play a central role in calculus and its applications. DEFINITION OF A FUNCTION Many scientic laws and engineering principles describe how one quantity depends on another. This idea was formalized in 1673 by Gottfried Wilhelm Leibniz (see p. xx) who coined the term function to indicate the dependence of one quantity on another, as described in the following denition. 0.1.1 denition If a variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x. Four common methods for representing functions are: Numerically by tables Geometrically by graphs Algebraically by formulas Verbally

August 31, 2011 19:31 C00 Sheet number 2 Page number 2 cyan magenta yellow black 2 Chapter 0 / Before Calculus The method of representation often depends on how the function arises. For example: Table 0.1.1 shows the top qualifying speed S for the Indianapolis 500 auto race as a Table 0.1.1 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 228.011 231.604 233.100 218.263 223.503 225.179 223.471 226.037 231.342 231.725 222.024 227.598 228.985 225.817 226.366 224.864 227.970 227.472 year t speed S (mi/h) indianapolis 500 qualifying speeds function of the year t. There is exactly one value of S for each value of t. Figure 0.1.1 is a graphical record of an earthquake recorded on a seismograph. The graph describes the deection D of the seismograph needle as a function of the time T elapsed since the wave left the earthquakes epicenter. There is exactly one value of D for each value of T . Some of the most familiar functions arise from formulas; for example, the formula C = 2r expresses the circumference C of a circle as a function of its radius r. There is exactly one value of C for each value of r. Sometimes functions are described in words. For example, Isaac Newtons Law of Universal Gravitation is often stated as follows: The gravitational force of attraction between two bodies in the Universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This is the verbal description of the formula F = G m1m2 r2 in which F is the force of attraction, m1 and m2 are the masses, r is the distance be- tween them, and G is a constant. If the masses are constant, then the verbal description denes F as a function of r. There is exactly one value of F for each value of r. T D 0 10 20 30 40 50 60 70 80 Time in minutes Time of earthquake shock Arrival of P-waves Arrival of S-waves 11.8 minutes 9.4 minutes Surface waves Figure 0.1.1 In the mid-eighteenth century the Swiss mathematician Leonhard Euler (pronounced oiler) conceived the idea of denoting functions by letters of the alphabet, thereby making it possible to refer to functions without stating specic formulas, graphs, or tables. To understand Eulers idea, think of a function as a computer program that takes an input x, operates on it in some way, and produces exactly one output y. The computer program is an object in its own right, so we can give it a name, say f . Thus, the function f (the computer program) associates a unique output y with each input x (Figure 0.1.2). This suggests the Input x Output y Computer Program f Figure 0.1.2 following denition. 0.1.2 denition A function f is a rule that associates a unique output with each input. If the input is denoted by x, then the output is denoted by f (x) (read f of x). In this denition the term unique means exactly one. Thus, a function cannot assign two different outputs to the same input. For example, Figure 0.1.3 shows a plot of weight WeightW(pounds) Age A (years) 10 15 20 25 30 50 75 100 125 150 175 200 225 Figure 0.1.3 versus age for a random sample of 100 college students. This plot does not describe W as a function of A because there are some values of A with more than one corresponding

August 31, 2011 19:31 C00 Sheet number 3 Page number 3 cyan magenta yellow black 0.1 Functions 3 value of W. This is to be expected, since two people with the same age can have different weights. INDEPENDENT AND DEPENDENT VARIABLES For a given input x, the output of a function f is called the value of f at x or the image of x under f . Sometimes we will want to denote the output by a single letter, say y, and write y = f(x) This equation expresses y as a function of x; the variable x is called the independent variable (or argument) of f , and the variable y is called the dependent variable of f . This terminology is intended to suggest that x is free to vary, but that once x has a specic value a corresponding value of y is determined. For now we will only consider functions in which the independent and dependent variables are real numbers, in which case we say that f is a real-valued function of a real variable. Later, we will consider other kinds of functions. Example 1 Table 0.1.2 describes a functional relationship y = f (x) for whichTable 0.1.2 0 3 x y 3 6 1 4 2 1 f(0) = 3 f associates y = 3 with x = 0. f(1) = 4 f associates y = 4 with x = 1. f(2) = 1 f associates y = 1 with x = 2. f(3) = 6 f associates y = 6 with x = 3. Example 2 The equation y = 3x2 4x + 2 has the form y = f(x) in which the function f is given by the formula f(x) = 3x2 4x + 2 Leonhard Euler (17071783) Euler was probably the most prolic mathematician who ever lived. It has been said that Euler wrote mathematics as effortlessly as most men breathe. He was born in Basel, Switzerland, and was the son of a Protestant minister who had himself studied mathematics. Eulers genius developed early. He attended the University of Basel, where by age 16 he obtained both a Bachelor ofArts degree and a Masters degree in philosophy. While at Basel, Euler had the good fortune to be tutored one day a week in mathematics by a distinguished mathematician, Johann Bernoulli. At the urging of his father, Euler then began to study theology. The lure of mathematics was too great, however, and by age 18 Euler had begun to do mathematical research. Nevertheless, the inuence of his father and his theological studies remained, and throughout his life Euler was a deeply religious, unaffected person. At various times Euler taught at St. Petersburg Academy of Sciences (in Rus- sia), the University of Basel, and the Berlin Academy of Sciences. Eulers energy and capacity for work were virtually boundless. His collected works form more than 100 quarto-sized volumes and it is believed that much of his work has been lost. What is particularly astonishing is that Euler was blind for the last 17 years of his life, and this was one of his most productive periods! Eulers awless memory was phenomenal. Early in his life he memorized the entire Aeneid by Virgil, and at age 70 he could not only recite the entire work but could also state the rst and last sentence on each page of the book from which he memorized the work. His ability to solve problems in his head was beyond belief. He worked out in his head major problems of lunar motion that bafed Isaac Newton and once did a complicated calculation in his head to settle an argument between two students whose computations differed in the ftieth decimal place. Following the development of calculus by Leibniz and Newton, results in mathematics developed rapidly in a disorganized way. Eu- lers genius gave coherence to the mathematical landscape. He was the rst mathematician to bring the full power of calculus to bear on problems from physics. He made major contributions to virtu- ally every branch of mathematics as well as to the theory of optics, planetary motion, electricity, magnetism, and general mechanics. [Image: http://commons.wikimedia.org/wiki/File:Leonhard_Euler_by_Handmann_.png]

August 31, 2011 19:31 C00 Sheet number 4 Page number 4 cyan magenta yellow black 4 Chapter 0 / Before Calculus For each input x, the corresponding output y is obtained by substituting x in this formula. For example, f(0) = 3(0)2 4(0) + 2 = 2 f associates y = 2 with x = 0. f(1.7) = 3(1.7)2 4(1.7) + 2 = 17.47 f associates y = 17.47 with x = 1.7. f( 2 ) = 3( 2 )2 4 2 + 2 = 8 4 2 f associates y = 8 4 2 with x = 2. GRAPHS OF FUNCTIONS If f is a real-valued function of a real variable, then the graph of f in the xy-plane is dened to be the graph of the equation y = f(x). For example, the graph of the function f(x) = x is the graph of the equation y = x, shown in Figure 0.1.4. That gure also shows the graphs of some other basic functions that may already be familiar to you. In Appendix A we discuss techniques for graphing functions using graphing technology. Figure 0.1.4 shows only portions of the graphs. Where appropriate, and unless indicated otherwise, it is understood that graphs shown in this text extend indenitely beyond the boundaries of the displayed gure. 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 x y y = x y = x2 y = x3 y = 1/x 3 2 1 10 2 3 1 1 0 2 3 4 5 6 7 x y 8 6 4 2 20 4 6 8 8 6 4 2 2 0 4 6 8 x y 54321 10 2 3 4 5 4 3 2 1 1 0 2 3 4 x y y = xy = x 3 8 6 4 2 20 4 6 8 4 3 2 1 1 0 2 3 4 x y 01 1 2 3 4 5 6 7 8 9 4 3 2 1 0 1 2 3 4 x y Figure 0.1.4 Since x is imaginary for negative val- ues of x, there are no points on the graph of y = x in the region where x < 0. Graphs can provide valuable visual information about a function. For example, since the graph of a function f in the xy-plane is the graph of the equation y = f(x), the points on the graph of f are of the form (x, f(x)); that is, the y-coordinate of a point on the graph of f is the value of f at the corresponding x-coordinate (Figure 0.1.5). The values of x for which f(x) = 0 are the x-coordinates of the points where the graph of f intersects the x-axis (Figure 0.1.6). These values are called the zeros of f , the roots of f(x) = 0, or the x-intercepts of the graph of y = f(x). x y (x, f(x)) f(x) y = f(x) x Figure 0.1.5 The y-coordinate of a point on the graph of y = f(x) is the value of f at the corresponding x-coordinate. THE VERTICAL LINE TEST Not every curve in the xy-plane is the graph of a function. For example, consider the curve in Figure 0.1.7, which is cut at two distinct points, (a, b) and (a, c), by a vertical line. This curve cannot be the graph of y = f(x) for any function f ; otherwise, we would have f(a) = b and f(a) = c

August 31, 2011 19:31 C00 Sheet number 5 Page number 5 cyan magenta yellow black 0.1 Functions 5 which is impossible, since f cannot assign two different values to a. Thus, there is no x y y = f(x) x1 0 x2 x3 Figure 0.1.6 f has zeros at x1, 0, x2, and x3. function f whose graph is the given curve. This illustrates the following general result, which we will call the vertical line test. 0.1.3 theverticallinetest Acurveinthexy-planeisthegraphofsomefunction f if and only if no vertical line intersects the curve more than once. x y a (a, b) (a, c) Figure 0.1.7 This curve cannot be the graph of a function. Example 3 The graph of the equation x2 + y2 = 25 is a circle of radius 5 centered at the origin and hence there are vertical lines that cut the graph more than once (Figure 0.1.8). Thus this equation does not dene y as a function of x. 6 6 6 6 x y x2 + y2 = 25 Figure 0.1.8 THE ABSOLUTE VALUE FUNCTION Recall that the absolute value or magnitude of a real number x is dened by |x| = x, x 0 x, x < 0 The effect of taking the absolute value of a number is to strip away the minus sign if the Symbols such as +x and x are de- ceptive, since it is tempting to conclude that +x is positive and x is negative. However, this need not be so, since x itself can be positive or negative. For example, if x is negative, say x = 3, then x = 3 is positive and +x = 3 is negative. number is negative and to leave the number unchanged if it is nonnegative. Thus, |5| = 5, 4 7 = 4 7 , |0| = 0 A more detailed discussion of the properties of absolute value is given in Web Appendix F. However, for convenience we provide the following summary of its algebraic properties. 0.1.4 properties of absolute value If a and b are real numbers, then (a) |a| = |a| A number and its negative have the same absolute value. (b) |ab| = |a| |b| The absolute value of a product is the product of the absolute values. (c) |a/b| = |a|/|b|, b = 0 The absolute value of a ratio is the ratio of the absolute values. (d) |a + b| |a| + |b| The triangle inequality The graph of the function f(x) = |x| can be obtained by graphing the two parts of the equation y = x, x 0 x, x < 0 separately. Combining the two parts produces the V-shaped graph in Figure 0.1.9. Absolute values have important relationships to square roots. To see why this is so, recall from algebra that every positive real number x has two square roots, one positive and one negative. By denition, the symbol x denotes the positive square root of x. WARNING To denote the negative square root you must write x. For example, the positive square root of 9 is 9 = 3, whereas the negative square root of 9 is 9 = 3. (Do not make the mis- take of writing 9 = 3.) Care must be exercised in simplifying expressions of the form x2, since it is not always true that x2 = x. This equation is correct if x is nonnegative, but it is false if x is negative. For example, if x = 4, then x2 = (4)2 = 16 = 4 = x

August 31, 2011 19:31 C00 Sheet number 6 Page number 6 cyan magenta yellow black 6 Chapter 0 / Before Calculus A statement that is correct for all real values of x is x2 = |x| (1) 5 4 3 2 1 10 2 3 4 5 3 2 1 1 0 2 3 4 5 x y y = |x| Figure 0.1.9 TECH NOLOGY MASTERY Verify (1) by using a graphing utility to show that the equations y = x2 and y = |x| have the same graph. PIECEWISE-DEFINED FUNCTIONS The absolute value function f(x) = |x| is an example of a function that is dened piecewise in the sense that the formula for f changes, depending on the value of x. Example 4 Sketch the graph of the function dened piecewise by the formula f(x) = 0, x 1 1 x2, 1 < x < 1 x, x 1 Solution. The formula for f changes at the points x = 1 and x = 1. (We call these the breakpoints for the formula.) A good procedure for graphing functions dened piecewise is to graph the function separately over the open intervals determined by the breakpoints, and then graph f at the breakpoints themselves. For the function f in this example the graph is the horizontal ray y = 0 on the interval (, 1], it is the semicircle y = 1 x2 on the interval (1, 1), and it is the ray y = x on the interval [1, +). The formula for f species that the equation y = 0 applies at the breakpoint 1 [so y = f(1) = 0], and it species that the equation y = x applies at the breakpoint 1 [so y = f(1) = 1]. The graph of f is shown in Figure 0.1.10. x y 12 1 2 1 2 Figure 0.1.10 REMARK In Figure 0.1.10 the solid dot and open circle at the breakpoint x = 1 serve to emphasize that the point on the graph lies on the ray and not the semicircle. There is no ambiguity at the breakpoint x = 1 because the two parts of the graph join together continuously there. Example 5 Increasing the speed at which air moves over a persons skin increases The wind chill index measures the sensation of coldness that we feel from the combined effect of temperature and wind speed. Brian Horisk/Alamy the rate of moisture evaporation and makes the person feel cooler. (This is why we fan ourselves in hot weather.) The wind chill index is the temperature at a wind speed of 4 mi/h that would produce the same sensation on exposed skin as the current temperature and wind speed combination. An empirical formula (i.e., a formula based on experimental data) for the wind chill index W at 32 F for a wind speed of v mi/h is W = 32, 0 v 3 55.628 22.07v0.16 , 3 < v A computer-generated graph of W(v) is shown in Figure 0.1.11. Figure 0.1.11 Wind chill versus wind speed at 32 F 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 5 0 10 15 20 25 30 35 Wind speed v (mi/h) WindchillW(F)

August 31, 2011 19:31 C00 Sheet number 7 Page number 7 cyan magenta yellow black 0.1 Functions 7 DOMAIN AND RANGE If x and y are related by the equation y = f(x), then the set of all allowable inputs (x-values) is called the domain of f , and the set of outputs (y-values) that result when x varies over the domain is called the range of f . For example, if f is the function dened by the table in Example 1, then the domain is the set {0, 1, 2, 3} and the range is the set {1, 3, 4, 6}. Sometimes physical or geometric considerations impose restrictions on the allowable inputs of a function. For example, if y denotes the area of a square of side x, then these variables are related by the equation y = x2 . Although this equation produces a unique value of y for every real number x, the fact that lengths must be nonnegative imposes the requirement that x 0. One might argue that a physical square cannot have a side of length zero. However, it is often convenient mathe- matically to allow zero lengths, and we will do so throughout this text where appropriate. When a function is dened by a mathematical formula, the formula itself may impose restrictions on the allowable inputs. For example, if y = 1/x, then x = 0 is not an allowable input since division by zero is undened, and if y = x, then negative values of x are not allowable inputs because they produce imaginary values for y and we have agreed to consider only real-valued functions of a real variable. In general, we make the following denition. 0.1.5 denition If a real-valued function of a real variable is dened by a formula, and if no domain is stated explicitly, then it is to be understood that the domain consists of all real numbers for which the formula yields a real value. This is called the natural domain of the function. The domain and range of a function f can be pictured by projecting the graph of y = f(x) onto the coordinate axes as shown in Figure 0.1.12. x y y = f(x) Domain Range Figure 0.1.12 The projection of y = f(x) on the x-axis is the set of allowable x-values for f , and the projection on the y-axis is the set of corresponding y-values. Example 6 Find the natural domain of (a) f(x) = x3 (b) f(x) = 1/[(x 1)(x 3)] (c) f(x) = tan x (d) f(x) = x2 5x + 6 Solution (a). The function f has real values for all real x, so its natural domain is the interval (, +). Solution (b). The function f has real values for all real x, except x = 1 and x = 3, where divisions by zero occur. Thus, the natural domain is {x : x = 1 and x = 3} = (, 1) (1, 3) (3, +) Solution (c). Since f(x) = tan x = sin x/ cos x, the function f has real values except where cos x = 0, and this occurs when x is an odd integer multiple of /2. Thus, the natural domain consists of all real numbers except For a review of trigonometry see Ap- pendix B. x = 2 , 3 2 , 5 2 , . . . Solution (d). The function f has real values, except when the expression inside the radical is negative. Thus the natural domain consists of all real numbers x such that x2 5x + 6 = (x 3)(x 2) 0 This inequality is satised if x 2 or x 3 (verify), so the natural domain of f is (, 2] [3, +)

August 31, 2011 19:31 C00 Sheet number 8 Page number 8 cyan magenta yellow black 8 Chapter 0 / Before Calculus In some cases we will state the domain explicitly when dening a function. For example, if f(x) = x2 is the area of a square of side x, then we can write f(x) = x2 , x 0 to indicate that we take the domain of f to be the set of nonnegative real numbers (Fig- ure 0.1.13).x y y = x2 x y y = x2 , x 0 Figure 0.1.13 THE EFFECT OF ALGEBRAIC OPERATIONS ON THE DOMAIN Algebraic expressions are frequently simplied by canceling common factors in the nu- merator and denominator. However, care must be exercised when simplifying formulas for functions in this way, since this process can alter the domain. Example 7 The natural domain of the function f(x) = x2 4 x 2 (2) consists of all real x except x = 2. However, if we factor the numerator and then cancel the common factor in the numerator and denominator, we obtain f(x) = (x 2)(x + 2) x 2 = x + 2 (3) Since the right side of (3) has a value of f (2) = 4 and f (2) was undened in (2), the algebraic simplication has changed the function. Geometrically, the graph of (3) is the line in Figure 0.1.14a, whereas the graph of (2) is the same line but with a hole at x = 2, since the function is undened there (Figure 0.1.14b). In short, the geometric effect of the algebraic cancellation is to eliminate the hole in the original graph. 321 1 2 3 4 5 1 2 3 4 5 6 x y y = x + 2 321 1 2 3 4 5 1 2 3 4 5 6 x y y = x 2 x2 4 (b) (a) Figure 0.1.14 Sometimesalterationstothedomainofafunctionthatresultfromalgebraicsimplication are irrelevant to the problem at hand and can be ignored. However, if the domain must be preserved, then one must impose the restrictions on the simplied function explicitly. For example, if we wanted to preserve the domain of the function in Example 7, then we would have to express the simplied form of the function as f(x) = x + 2, x = 2 Example 8 Find the domain and range of (a) f(x) = 2 + x 1 (b) f(x) = (x + 1)/(x 1) Solution (a). Since no domain is stated explicitly, the domain of f is its natural domain, [1, +). As x varies over the interval [1, +), the value of x 1 varies over the interval [0, +), so the value of f(x) = 2 + x 1 varies over the interval [2, +), which is the range of f . The domain and range are highlighted in green on the x- and y-axes in Figure 0.1.15. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 y y = 2 + x 1 x Figure 0.1.15 Solution (b). The given function f is dened for all real x, except x = 1, so the natural domain of f is {x : x = 1} = (, 1) (1, +)

August 31, 2011 19:31 C00 Sheet number 9 Page number 9 cyan magenta yellow black 0.1 Functions 9 To determine the range it will be convenient to introduce a dependent variable y = x + 1 x 1 (4) Although the set of possible y-values is not immediately evident from this equation, the graph of (4), which is shown in Figure 0.1.16, suggests that the range of f consists of all y, except y = 1. To see that this is so, we solve (4) for x in terms of y: (x 1)y = x + 1 xy y = x + 1 xy x = y + 1 x(y 1) = y + 1 x = y + 1 y 1 It is now evident from the right side of this equation that y = 1 is not in the range; otherwise we would have a division by zero. No other values of y are excluded by this equation, so the range of the function f is {y : y = 1} = (, 1) (1, +), which agrees with the result obtained graphically. 3 2 1 1 2 3 4 5 6 2 1 1 2 3 4 5 y y = x 1 x + 1 x Figure 0.1.16 DOMAIN AND RANGE IN APPLIED PROBLEMS In applications, physical considerations often impose restrictions on the domain and range of a function. Example 9 An open box is to be made from a 16-inch by 30-inch piece of card- board by cutting out squares of equal size from the four corners and bending up the sides (Figure 0.1.17a). (a) Let V be the volume of the box that results when the squares have sides of length x. Find a formula for V as a function of x. (b) Find the domain of V . (c) Use the graph of V given in Figure 0.1.17c to estimate the range of V . (d) Describe in words what the graph tells you about the volume. Solution (a). As shown in Figure 0.1.17b, the resulting box has dimensions 16 2x by 30 2x by x, so the volume V (x) is given by V (x) = (16 2x)(30 2x)x = 480x 92x2 + 4x3 0 1 2 3 4 5 6 7 8 9 100 200 300 400 500 600 700 800 VolumeVofbox(in3 ) Side x of square cut (in) x x x x x x x x 16 in 30 in 16 2x 30 2x x (c)(b)(a) Figure 0.1.17

August 31, 2011 19:31 C00 Sheet number 10 Page number 10 cyan magenta yellow black 10 Chapter 0 / Before Calculus Solution (b). The domain is the set of x-values and the range is the set of V -values. Because x is a length, it must be nonnegative, and because we cannot cut out squares whose sides are more than 8 in long (why?), the x-values in the domain must satisfy 0 x 8 Solution (c). From the graph of V versus x in Figure 0.1.17c we estimate that the V - values in the range satisfy 0 V 725 Note that this is an approximation. Later we will show how to nd the range exactly. Solution (d). The graph tells us that the box of maximum volume occurs for a value of x that is between 3 and 4 and that the maximum volume is approximately 725 in3 . The graph also shows that the volume decreases toward zero as x gets closer to 0 or 8, which should make sense to you intuitively. In applications involving time, formulas for functions are often expressed in terms of a variable t whose starting value is taken to be t = 0. Example 10 At 8:05 A.M. a car is clocked at 100 ft/s by a radar detector that is positioned at the edge of a straight highway. Assuming that the car maintains a constant speed between 8:05 A.M. and 8:06 A.M., nd a function D(t) that expresses the distance traveled by the car during that time interval as a function of the time t. Solution. It would be clumsy to use the actual clock time for the variable t, so let us agree to use the elapsed time in seconds, starting with t = 0 at 8:05 A.M. and ending with t = 60 at 8:06 A.M. At each instant, the distance traveled (in ft) is equal to the speed of the car (in ft/s) multiplied by the elapsed time (in s). Thus, D(t) = 100t, 0 t 60 The graph of D versus t is shown in Figure 0.1.18. 0 10 20 30 40 50 60 1000 2000 3000 4000 5000 6000 Radar Tracking Time t (s)8:05 a.m. 8:06 a.m. DistanceD(ft) Figure 0.1.18 ISSUES OF SCALE AND UNITS In geometric problems where you want to preserve the true shape of a graph, you must use units of equal length on both axes. For example, if you graph a circle in a coordinate system in which 1 unit in the y-direction is smaller than 1 unit in the x-direction, then the circle will be squashed vertically into an elliptical shape (Figure 0.1.19). x y The circle is squashed because 1 unit on the y-axis has a smaller length than 1 unit on the x-axis. Figure 0.1.19 In applications where the variables on the two axes have unrelated units (say, centimeters on the y-axis and seconds on the x-axis), then nothing is gained by requiring the units to have equal lengths; choose the lengths to make the graph as clear as possible. However, sometimes it is inconvenient or impossible to display a graph using units of equal length. For example, consider the equation y = x2 If we want to show the portion of the graph over the interval 3 x 3, then there is no problem using units of equal length, since y only varies from 0 to 9 over that interval. However, if we want to show the portion of the graph over the interval 10 x 10, then there is a problem keeping the units equal in length, since the value of y varies between 0 and 100. In this case the only reasonable way to show all of the graph that occurs over the interval 10 x 10 is to compress the unit of length along the y-axis, as illustrated in Figure 0.1.20.

August 31, 2011 19:31 C00 Sheet number 11 Page number 11 cyan magenta yellow black 0.1 Functions 11 Figure 0.1.20 3 2 1 1 2 3 1 2 3 4 5 6 7 8 9 x y 10 5 5 10 20 40 60 80 100 x y QUICK CHECK EXERCISES 0.1 (See page 15 for answers.) 1. Let f(x) = x + 1 + 4. (a) The natural domain of f is . (b) f(3) = (c) f (t2 1) = (d) f(x) = 7 if x = (e) The range of f is . 2. Line segments in an xy-plane form letters as depicted. (a) If the y-axis is parallel to the letter I, which of the letters represent the graph of y = f(x) for some function f ? (b) If the y-axis is perpendicular to the letter I, which of the letters represent the graph of y = f(x) for some function f ? 3. The accompanying gure shows the complete graph of y = f(x). (a) The domain of f is . (b) The range of f is . (c) f (3) = (d) f 1 2 = (e) The solutions to f(x) = 3 2 are x = and x = . 3 2 1 321 2 1 1 2 x y Figure Ex-3 4. The accompanying table gives a 5-day forecast of high and low temperatures in degrees Fahrenheit ( F). (a) Suppose that x and y denote the respective high and low temperature predictions for each of the 5 days. Is y a function of x? If so, give the domain and range of this function. (b) Suppose that x and y denote the respective low and high temperature predictions for each of the 5 days. Is y a function of x? If so, give the domain and range of this function. 75 52 high low 70 50 71 56 65 48 73 52 mon tue wed thurs fri Table Ex-4 5. Let l, w, and A denote the length, width, and area of a rectangle, respectively, and suppose that the width of the rectangle is half the length. (a) If l is expressed as a function of w, then l = . (b) If A is expressed as a function of l, then A = . (c) If w is expressed as a function of A, then w = .

August 31, 2011 19:31 C00 Sheet number 12 Page number 12 cyan magenta yellow black 12 Chapter 0 / Before Calculus EXERCISE SET 0.1 Graphing Utility 1. Use the accompanying graph to answer the following ques- tions, making reasonable approximations where needed. (a) For what values of x is y = 1? (b) For what values of x is y = 3? (c) For what values of y is x = 3? (d) For what values of x is y 0? (e) What are the maximum and minimum values of y and for what values of x do they occur? 3 2 1 10 2 3 3 2 1 1 0 2 3 x y Figure Ex-1 2. Use the accompanying table to answer the questions posed in Exercise 1. 2 5 x y 2 7 1 1 0 2 3 1 4 1 5 0 6 9 Table Ex-2 3. In each part of the accompanying gure, determine whether the graph denes y as a function of x. x y (c) x y (d) x y (b) x y (a) Figure Ex-3 4. In each part, compare the natural domains of f and g. (a) f(x) = x2 + x x + 1 ; g(x) = x (b) f(x) = x x + x x + 1 ; g(x) = x FOCUS ON CONCEPTS 5. The accompanying graph shows the median income in U.S. households (adjusted for ination) between 1990 and 2005. Use the graph to answer the following ques- tions, making reasonable approximations where needed. (a) When was the median income at its maximum value, and what was the median income when that occurred? (b) When was the median income at its minimum value, and what was the median income when that occurred? (c) The median income was declining during the 2-year period between 2000 and 2002. Was it declining more rapidly during the rst year or the second year of that period? Explain your reasoning. 1990 1995 2000 2005 42 46 44 48 Median U.S. Household Income in Thousands of Constant 2005 Dollars MedianU.S.HouseholdIncome Source: U.S. Census Bureau, August 2006. Figure Ex-5 6. Use the median income graph in Exercise 5 to answer the following questions, making reasonable approximations where needed. (a) What was the average yearly growth of median in- come between 1993 and 1999? (b) The median income was increasing during the 6-year period between 1993 and 1999. Was it increasing more rapidly during the rst 3 years or the last 3 years of that period? Explain your reasoning. (c) Consider the statement: After years of decline, me- dian income this year was nally higher than that of last year. In what years would this statement have been correct?

August 31, 2011 19:31 C00 Sheet number 13 Page number 13 cyan magenta yellow black 0.1 Functions 13 7. Find f(0), f(2), f(2), f(3), f( 2 ), and f(3t). (a) f(x) = 3x2 2 (b) f(x) = 1 x , x > 3 2x, x 3 8. Find g(3), g(1), g(), g(1.1), and g(t2 1). (a) g(x) = x + 1 x 1 (b) g(x) = x + 1, x 1 3, x < 1 910 Find the natural domain and determine the range of each function. If you have a graphing utility, use it to conrm that your result is consistent with the graph produced by your graph- ing utility. [Note: Set your graphing utility in radian mode when graphing trigonometric functions.] 9. (a) f(x) = 1 x 3 (b) F(x) = x |x| (c) g(x) = x2 3 (d) G(x) = x2 2x + 5 (e) h(x) = 1 1 sin x (f ) H(x) = x2 4 x 2 10. (a) f(x) = 3 x (b) F(x) = 4 x2 (c) g(x) = 3 + x (d) G(x) = x3 + 2 (e) h(x) = 3 sin x (f ) H(x) = (sin x)2 FOCUS ON CONCEPTS 11. (a) If you had a device that could record the Earths pop- ulation continuously, would you expect the graph of population versus time to be a continuous (unbro- ken) curve? Explain what might cause breaks in the curve. (b) Suppose that a hospital patient receives an injection of an antibiotic every 8 hours and that between in- jections the concentration C of the antibiotic in the bloodstream decreases as the antibiotic is absorbed by the tissues. What might the graph of C versus the elapsed time t look like? 12. (a) If you had a device that could record the tempera- ture of a room continuously over a 24-hour period, would you expect the graph of temperature versus time to be a continuous (unbroken) curve? Explain your reasoning. (b) If you had a computer that could track the number of boxes of cereal on the shelf of a market contin- uously over a 1-week period, would you expect the graph of the number of boxes on the shelf versus time to be a continuous (unbroken) curve? Explain your reasoning. 13. A boat is bobbing up and down on some gentle waves. Suddenly it gets hit by a large wave and sinks. Sketch a rough graph of the height of the boat above the ocean oor as a function of time. 14. A cup of hot coffee sits on a table. You pour in some cool milk and let it sit for an hour. Sketch a rough graph of the temperature of the coffee as a function of time. 1518 As seen in Example 3, the equation x2 + y2 = 25 does not dene y as a function of x. Each graph in these exercises is a portion of the circle x2 + y2 = 25. In each case, determine whether the graph denes y as a function of x, and if so, give a formula for y in terms of x. 15. 5 5 5 5 x y 16. 5 5 5 5 x y 17. 5 5 5 5 x y 18. 5 5 5 5 x y 1922 TrueFalse Determine whether the statement is true or false. Explain your answer. 19. A curve that crosses the x-axis at two different points cannot be the graph of a function. 20. The natural domain of a real-valued function dened by a formula consists of all those real numbers for which the formula yields a real value. 21. The range of the absolute value function is all positive real numbers. 22. If g(x) = 1/ f(x), then the domain of g consists of all those real numbers x for which f(x) = 0. 23. Use the equation y = x2 6x + 8 to answer the following questions. (a) For what values of x is y = 0? (b) For what values of x is y = 10? (c) For what values of x is y 0? (d) Does y have a minimum value? A maximum value? If so, nd them. 24. Use the equation y = 1 + x to answer the following ques- tions. (a) For what values of x is y = 4? (b) For what values of x is y = 0? (c) For what values of x is y 6? (cont.)

August 31, 2011 19:31 C00 Sheet number 14 Page number 14 cyan magenta yellow black 14 Chapter 0 / Before Calculus (d) Does y have a minimum value? A maximum value? If so, nd them. 25. As shown in the accompanying gure, a pendulum of con- stant length L makes an angle with its vertical position. Express the height h as a function of the angle . 26. Express the length L of a chord of a circle with radius 10 cm as a function of the central angle (see the accompanying gure). L h u Figure Ex-25 L 10 cm u Figure Ex-26 2728 Express the function in piecewise form without using absolute values. [Suggestion: It may help to generate the graph of the function.] 27. (a) f(x) = |x| + 3x + 1 (b) g(x) = |x| + |x 1| 28. (a) f(x) = 3 + |2x 5| (b) g(x) = 3|x 2| |x + 1| 29. As shown in the accompanying gure, an open box is to be constructed from a rectangular sheet of metal, 8 in by 15 in, by cutting out squares with sides of length x from each corner and bending up the sides. (a) Express the volume V as a function of x. (b) Find the domain of V . (c) Plot the graph of the function V obtained in part (a) and estimate the range of this function. (d) In words, describe how the volume V varies with x, and discuss how one might construct boxes of maximum volume. x x x x x x x x 8 in 15 in Figure Ex-29 30. Repeat Exercise 29 assuming the box is constructed in the same fashion from a 6-inch-square sheet of metal. 31. A construction company has adjoined a 1000 ft2 rectan- gular enclosure to its ofce building. Three sides of the enclosure are fenced in. The side of the building adjacent to the enclosure is 100 ft long and a portion of this side is used as the fourth side of the enclosure. Let x and y be the dimensions of the enclosure, where x is measured parallel to the building, and let L be the length of fencing required for those dimensions. (a) Find a formula for L in terms of x and y. (b) Find a formula that expresses L as a function of x alone. (c) What is the domain of the function in part (b)? (d) Plot the function in part (b) and estimate the dimensions of the enclosure that minimize the amount of fencing required. 32. As shown in the accompanying gure, a camera is mounted at a point 3000 ft from the base of a rocket launching pad. The rocket rises vertically when launched, and the cameras elevation angle is continually adjusted to follow the bottom of the rocket. (a) Express the height x as a function of the elevation an- gle . (b) Find the domain of the function in part (a). (c) Plot the graph of the function in part (a) and use it to estimate the height of the rocket when the elevation an- gle is /4 0.7854 radian. Compare this estimate to the exact height. 3000 ft x Camera Rocket u Figure Ex-32 33. A soup company wants to manufacture a can in the shape of a right circular cylinder that will hold 500 cm3 of liquid. The material for the top and bottom costs 0.02 cent/cm2 , and the material for the sides costs 0.01 cent/cm2 . (a) Estimate the radius r and the height h of the can that costs the least to manufacture. [Suggestion: Express the cost C in terms of r.] (b) Suppose that the tops and bottoms of radius r are punched out from square sheets with sides of length 2r and the scraps are waste. If you allow for the cost of the waste, would you expect the can of least cost to be taller or shorter than the one in part (a)? Explain. (c) Estimate the radius, height, and cost of the can in part (b), and determine whether your conjecture was correct. 34. The designer of a sports facility wants to put a quarter-mile (1320 ft) running track around a football eld, oriented as in the accompanying gure on the next page. The football eld is 360 ft long (including the end zones) and 160 ft wide. The track consists of two straightaways and two semicircles, with the straightaways extending at least the length of the football eld. (a) Show that it is possible to construct a quarter-mile track around the football eld. [Suggestion: Find the shortest track that can be constructed around the eld.] (b) Let L be the length of a straightaway (in feet), and let x be the distance (in feet) between a sideline of the foot- ball eld and a straightaway. Make a graph of L ver- sus x. (cont.)

August 31, 2011 19:31 C00 Sheet number 15 Page number 15 cyan magenta yellow black 0.2 New Functions from Old 15 (c) Use the graph to estimate the value of x that produces the shortest straightaways, and then nd this value of x exactly. (d) Use the graph to estimate the length of the longest pos- sible straightaways, and then nd that length exactly. 360 160 Figure Ex-34 3536 (i) Explain why the function f has one or more holes in its graph, and state the x-values at which those holes occur. (ii) Find a function g whose graph is identical to that of f, but without the holes. 35. f(x) = (x + 2)(x2 1) (x + 2)(x 1) 36. f(x) = x2 + |x| |x| 37. In 2001 the NationalWeather Service introduced a new wind chill temperature (WCT) index. For a given outside temper- ature T and wind speed v, the wind chill temperature index is the equivalent temperature that exposed skin would feel with a wind speed of v mi/h. Based on a more accurate model of cooling due to wind, the new formula is WCT = T, 0 v 3 35.74 + 0.6215T 35.75v0.16 + 0.4275T v0.16, 3 < v where T is the temperature in F, v is the wind speed in mi/h, and WCT is the equivalent temperature in F. Find the WCT to the nearest degree if T = 25 F and (a) v = 3 mi/h (b) v = 15 mi/h (c) v = 46 mi/h. Source: Adapted from UMAP Module 658, Windchill, W. Bosch and L. Cobb, COMAP, Arlington, MA. 3840 Use the formula for the wind chill temperature index described in Exercise 37. 38. Find the air temperature to the nearest degree if the WCT is reported as 60 F with a wind speed of 48 mi/h. 39. Find the air temperature to the nearest degree if the WCT is reported as 10 F with a wind speed of 48 mi/h. 40. Find the wind speed to the nearest mile per hour if the WCT is reported as 5 F with an air temperature of 20 F. QUICK CHECK ANSWERS 0.1 1. (a) [1, +) (b) 6 (c) |t| + 4 (d) 8 (e) [4, +) 2. (a) M (b) I 3. (a) [3, 3) (b) [2, 2] (c) 1 (d) 1 (e) 3 4 ; 3 2 4. (a) yes; domain: {65, 70, 71, 73, 75}; range: {48, 50, 52, 56} (b) no 5. (a) l = 2w (b) A = l2/2 (c) w = A/2 0.2 NEW FUNCTIONS FROM OLD Just as numbers can be added, subtracted, multiplied, and divided to produce other numbers, so functions can be added, subtracted, multiplied, and divided to produce other functions. In this section we will discuss these operations and some others that have no analogs in ordinary arithmetic. ARITHMETIC OPERATIONS ON FUNCTIONS Two functions, f and g, can be added, subtracted, multiplied, and divided in a natural way to form new functions f + g, f g, fg, and f /g. For example, f + g is dened by the formula (f + g)(x) = f(x) + g(x) (1) which states that for each input the value of f + g is obtained by adding the values of f and g. Equation (1) provides a formula for f + g but does not say anything about the domain of f + g. However, for the right side of this equation to be dened, x must lie in the domains of both f and g, so we dene the domain of f + g to be the intersection of these two domains. More generally, we make the following denition.

August 31, 2011 19:31 C00 Sheet number 16 Page number 16 cyan magenta yellow black 16 Chapter 0 / Before Calculus 0.2.1 denition Given functions f and g, we dene (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (fg)(x) = f(x)g(x) (f /g)(x) = f(x)/g(x) For the functions f + g, f g, and fg we dene the domain to be the intersection of the domains of f and g, and for the function f /g we dene the domain to be the intersection of the domains of f and g but with the points where g(x) = 0 excluded (to avoid divis